Prove that ALL is equivalent to EQUIV under mapping reductions, where:
ALL = { < M > | Turing machine M accepts all strings }
EQUIV = { < M_{1}, M_{2} > | M_{1} & M_{2} are Turing machines and L(M_{1}) = L(M_{2}) }
Prove directly that A_{TM} mapping reduces to FINITE and the complement of A_{TM} mapping reduces to FINITE, where:
FINITE = { < M > | L(M) contains a finite number of strings }
FAS = { (G,k) | G = (V,E) is a directed graph, k <= |E|, and there exists a subset E' of E such that |E'| <= k and E' contains at least one edge from each cycle in G. }Note: FAS = Feedback Arc Set. Also, there can be exponentially many cycles in a directed graph, so be careful when you show that FAS is in NP.
Optional: Show that the following language is NP-complete (instead of FAS).
Technically, Set_Basis is the language consisting of the triples (C, S, K) such that C has a set basis (as defined above) of size K. Hint: reduce from Vertex Cover. Second Hint: you can make the reduction work by constructing a collection C where each element of C has <= 3 elements from S.
NP is not equal to SPACE[n^{9}].